MIT OpenCourseWare http://ocw.mit.edu HST.583 Functional Magnetic Resonance Imaging: Data Acquisition and Analysis Fall 2006 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
HST.583: Functional Magnetic Resonance Imaging: Data Acquisition and Analysis, Fall 2006 Harvard-MIT Division of Health Sciences and Technology Course Director: Dr. Randy Gollub. Measuring Water Diffusion In Biological Systems Using Nuclear Magnetic Resonance Image removed due to copyright restrictions. "Diffusion-weighted axial image" http://www.medicineau.net.au/clinical/radiology/radiolog1768.html Karl Helmer HST 583, 2006
Why Would We Want to Measure the Self - Diffusion Coefficient of Water In Biological Tissue?
Why Would We Want to Measure the Self - Diffusion Coefficient of Water In Biological Tissue? We Don t.
Why Would We Want to Measure the Self - Diffusion Coefficient of Water In Biological Tissue? We Don t. What we are really interested in is how what we measure for a diffusion-weighted signal reflects the structure of the sample.
Why Would We Want to Measure the Self - Diffusion Coefficient of Water? We Don t. What we are really interested in is how what we measure for a diffusion-weighted signal reflects the structure of the sample. So, what are we measuring???
How Can the Diffusion Coefficient Reflect Sample Structure? Self-diffusion in bulk samples is a wellunderstood random process - Displacement (z) has a Gaussian probability distribution <z 2 > 1/2 = (2nDt) 1/2 probability(t) Courtesy of InductiveLoad. D = Self-Diffusion Coefficient n = # of dimensions z
How Can We Measure the Diffusion Coefficient of Water Using NMR?
How Can We Measure the Diffusion Coefficient of Water Using NMR? We Can t.
How Can We Measure the Diffusion Coefficient of Water Using NMR? We Can t. Instead we measure the displacement of the ensemble of spins in our sample and infer the diffusion coefficient.
How can we measures the (mean) displacement of water molecules using NMR? g(z) is a magnetic field added to B 0 that varies with position. ω(z) = γ (B 0 + g(z) z)
How can we measures the (mean) displacement of water molecules using NMR? z = 0 Tagging the initial position using phase of M z Applying g(z) for a time δ results in a phase shift that depends upon location in z
Now, after waiting a time we apply an equal gradient, but with the opposite sign Apply -g(z) for a time δ z if no diffusion: signal = M 0
But, in reality, there is always diffusion so we find that: Apply -g(z) for a time δ z if diffusion: signal = M 0 e (-q2 Dt) (t = - δ/3) q = q(g)
DW Spin Echo Pulse Sequences π/2 π δ Δ δ = gradient duration Δ = separation of gradient leading edges
But what do we do with: signal = M = M 0 e (-q2 Dt)? One equation, but two unknowns (M 0, D) How do we get another equation?
Change the diffusion-sensitizing gradient to a different value and acquire more data. q 2 t b = q 2 t = 0 ln(m) Slope = D Intercept = ln(m 0 ) b = q 2 t 0
Unrestricted Diffusion r' r
Restricted Diffusion r r'
The effect of barriers to the free diffusion of water molecules is to modify their probability distribution. P(z) Diffusion coefficient decreases with increasing diffusion time
Determination of D? Slope = D t dif ln(m/m0) 0-1 -2-3 -4 bead pack water -5-6 bulk water Slope = D 0 t dif -7 0.0 0.5 1.0 1.5 q 2 x 10 7 [1/cm 2 ] a = 15.8 μm bead pack, t dif = 50 ms, δ = 1.5 ms, g(max) = 72.8 G/cm See Helmer, et al. NMR in Biomedicine 8 (1995): 297-306.
Water Diffusion in an Ordered System High q 0-1 ln(m/m0) -2-3 -4-5 2π/a -6-7 0 1 2 3 4 5 6 q 2 k 2 x 10 7 [1/cm 2 ] a = 15.8 μm bead pack, t dif = 100 ms
Short diffusion times: Long diffusion times:
D (t dif ) gives information on different length scales 160 S/V T = tortuosity S/V = surface-to-volume ratio D (t) D(t) x 10-7 [cm 2 /sec] ] 120 80 40 0 1/T t 1/2 [sec 1/2 ] 0 0.2 0.4 0.6 0.8 1 a = 15.8 μm bead pack t
DW-Weighted Tumor Data ln M(q,t)/M(0,t) 0.0-0.5-1.0-1.5-2.0-2.5-3.0-3.5 0 50 100 q 2 [x10-9 m -2 ] 150 t dif = 42 ms 92 ms 192 ms 292 ms 492 ms D(t) Apparent Diffusion Coefficient (ADC)
ADC(t) for water in a RIF-1 Mouse Tumor D(t) 10 5 [cm 2 /s] Necrosis!! 0.10 0.24 0.60 0.75 0.10 2.55 (t) 1/2 [s 1/2 ]
ADC for water in a RIF-1 Mouse Tumor Control Day 1 Day 2 Day 3 Day 4 > 255 x10-7 ADC cm 2 /sec 1 x 10-7 Tumor Volume 0.68 cm 3 0.97 cm 3 1.26 cm 3 1.42 cm 3 Day 5 Day 6 Histology ADC Tumor Volume 1.70 cm 3 2.04 cm 3
ADC for water in a RIF-1 Mouse Tumor Treatment, 100mg/kg 5-FU Day 1 Day 2 Day 3 Day 4 Day 5 Day 6 ADC Tumor Volume 0.60 cm 3 0.70 cm 3 0.95 cm 3 0.86 cm 3 0.71 cm 3 0.76 cm 3 > 255 x10-7 cm 2 /sec 1 x 10-7 ADC Day 7 Day 8 Day 9 Day 10 Day 11 Histology > 255 x10-7 cm 2 /sec 1 x 10-7 Tumor Volume 1.13 cm 3 1.36 cm 3 1.60 cm 3 1.79 cm 3 2.08 cm 3
ADC av Maps vs Post-Occlusion Time Rat Brain 30 min Occlusion See Fuhai Li, M. D., K. Helmer, et al. "Secondary Decline in Apparent Diffusion Coefficient and Neurological Outcomes after a Short Period of Focal Brain Ischemia in Rats." Ann Neurol 48, no. 2 (2000): 236. MCAO 2 hr 3 hr 4 hr 5 hr 6 hr 7 hr 8 hr 9 hr 10 hr 11 hr 12 hr ADC (x10-5 mm 2 /s) ROI Positions < 30 > 60
80 ADC av Maps vs Post-Occlusion Time Rat Brain 30 min Occlusion Temporal ADC Changes in the Caudoputamen: 30-minute Transient Occlusion (n = 4) 75 70 ADC (x10-5 mm 2 /s) 65 60 55 50 45 40 35 30 Rep 1 2 3 4 5 6 7 8 9 10 11 12 Time (hours post reperfusion) Ipsilateral Contralateral See Fuhai Li, M. D., and K. Helmer, et al. "Secondary Decline in Apparent Diffusion Coefficient and Neurological Outcomes after a Short Period of Focal Brain Ischemia in Rats." Ann Neurol 48, no. 2 (2000): 236.
Issues with Interpreting DW Data In biological tissue, there are always restrictions. How then can we interpret the diffusion attenuation curve?
Biology-based Model: Intracellular and extracellular compartments D Biexponential Model with a distribution of cell sizes and shapes. Fast Exchange = f 1 D 1 + (1 f 1 ) D 2 S = S 0 ( f 1 e bd 1 + (1 f 1 ) e bd 2 ) Slow Exchange But real systems are rarely either/or.
DW-Weighted Tumor Data ln M(q,t)/M(0,t) 0.0-0.5-1.0-1.5-2.0-2.5-3.0-3.5 0 50 100 q 2 [x10-9 m -2 ] 150 t dif = 42 ms 92 ms 192 ms 292 ms 492 ms What does non-monexponentiality tell us?
Fast and Slow Diffusion? 0 Slope = D fast t dif ln(m/m0) -1-2 -3-4 -5-6 bulk water Slope = D slow t dif -7 0.0 0.5 1.0 1.5 q 2 x 10 7 [1/cm 2 ] See Helmer, et al. NMR in Biomedicine 8 (1995): 297-306.
Does Fast and Slow Mean Extracellular and Intracellular? No, because: 1)The same shape of curve can be found in the diffusion attenuation curve of single compartment systems (e.g., beads). 2) It gives almost exactly the opposite values for extra- and intracellular volume fractions (20/80 instead of 80/20 for IC/EC). Exchange?
What does fast and slow measure? Answer: It depends on range of b-values TE t dif sample structure sample tortuosity Fig 1 in Clark, C. A., et al. "In Vivo Mapping of the Fast and Slow Diffusion Tensors in Human Brain." Magn Reson Med 47, no. 4 (April 2002): 623-8. doi:10.1002/mrm.10118. Copyright (c) 2002 Wiley-Liss Inc. Reprinted with permission of John Wiley & Sons., Inc.
D ave (fast) D ave (slow) FA(fast) FA(slow) Fig 1 in Clark, C. A., et al. "In Vivo Mapping of the Fast and Slow Diffusion Tensors in Human Brain." Magn Reson Med 47, no. 4 (April 2002): 623-8. doi:10.1002/mrm.10118. Copyright (c) 2002 Wiley-Liss Inc. Reprinted with permission of John Wiley & Sons., Inc. slow restricted
Do We Get More Information by Using the Entire Diffusion Attenuation Curve? 0.0-0.5 ln M(q,t)/M(0,t) -1.0-1.5-2.0-2.5-3.0-3.5 0 50 100 q 2 [x10-9 m -2 ] 150
Practical Issues in DWI How do I choose my lowest b-value? 1)Diffusion gradients act like primer-crusher pairs. Therefore, slice profile of g = 0 image will be different from g 0 image. 2) Diffusion gradients also suppress flowing spins. Therefore, the use of a g = 0 image is discouraged.
Practical Issues in DWI How do I choose my highest b-value? 1. Greatest SNR in calculated ADC: bi D Ii = I0e I = true signal S i = σ = I i ε + ε 2 1/ 2 S = measured signal ε = noise
Practical Issues in DWI D = ln S ln b S 1 0, b = q 2 t σ 1 σ 2 2 2 2 D ( ) (1 2 bd D σ 2 0 + σ1 = + e 2 2 b b I0 ) SNR D = D σ D = (1 + bd e 0 2bD 1/ 2 I ) I σ F( bd) SNR 0
Practical Issues in DWI How do I choose my highest b-value? 2. Greatest sensitivity to %ΔADC: I D max bd =1.0
Practical Issues in DWI How to distribute the b-values? q 2 t This or? ln(m)
Practical Issues in DWI How to distribute the b-values? q 2 t This or? ln(m)
Practical Issues in DWI How to distribute the b-values? q 2 t This? ln(m)
Multiple measurements of 2 b-values are better than multiple different b-values. If the number of measurements can be large, then N high-b = N low-b 3.6 Note that depending on N and how you estimate the error, you can get different numbers for the optimum values, but Δb opt ~ 1(+)/D and N high-b ~ N low-b 4
Diffusion Tensor Imaging What effect does the direction of the diffusionsensitizing gradient have upon what we measure? y x In the 1- dimensional case (we measure D x or D y ): D y D 0, the bulk value D x <(<) D 0 D / ADC is a scalar
What effect does the direction of the diffusion-sensitizing gradient have upon what we measure? y z x In the 3- dimensional case (we measure D x,d y and D z ): D y D 0, the bulk value D x = D z <(<) D 0 D = (D x, D y, D z )
Diffusion Tensor Imaging Why not stick with vectors? Because is not z x y
The ADC is greatest along White Matter fiber tracts. Taylor et al., Biol Psychiatry, 55, 201 (2004) Courtesy Elsevier, Inc., http://www.sciencedirect.com. Used with permission.
1. There is nothing special about using tensors to characterize anisotropic diffusion. Rotate to principal frame to get eigenvalues.
Rotational Invariants for 3D Tensors. Table from: P.B. Kingsley, "Introduction to Diffusion Tensor Imaging Mathematics: Part I. Tensors, Rotations, and Eigenvectors." Concepts Magn Reson 28A no. 2 (2006): 101-122. Copyright (c) 2006 Wiley-Liss Inc. Reprinted with permission of John Wiley & Sons., Inc. Eigenvalues = D1, D2, D3 or λ 1, λ 2, λ 3 D av = (D xx + D yy + D zz )/3
Trace Imaging and b-value Strength Set of three images with caption removed due to copyright restrictions. Figure 1 in Maier, S. E., et al. "Normal Brain and Brain Tumor: Multicomponent Apparent Diffusion Coefficient Line Scan Imaging." Radiology 219 (2001): 842-849.
Distribution of Gradient Sampling Directions Need at least 6 different sampling directions Fig 2 image + caption, from: Le Bihan, D., et al. "Diffusion Tensor Imaging: Concepts and Applications." JMRI 13, no 4 (2001): 534-546. Copyright (c) 2001 Wiley-Liss, Inc., a subsidiary of John Wiley & Sons, Inc. Reprinted with permission of John Wiley & Sons., Inc.
Diffusion Tractography Follow Voxels With Largest Eigenvalues Being Continuous Between Two Regions of Interest Courtesy of Dr. Martha Shenton. Used with permission. Source: Shenton, M. E., M. Kubicki, and R. W. McCarley. "Diffusion Tensor Imaging: Image Acquisition and Processing Tools." SPL Technical Report 354, 2002.